A finite element method for the generalized Ericksen model of nematic liquid crystals

Abstract

We consider the generalized Ericksen model of liquid crystals, which is an energy with 8 independent “elastic”constants that depends on two order parameters n (director) and s (variable degree of orientation). In addition, we present a new finite element discretization for this energy, that can handle the degenerate elliptic part without regularization, with the following properties: it is stable and it Γ-converges to the continuous energy. Moreover, it does not require the mesh to be weakly acute (which was an important assumption in our previous work). Furthermore, we include other effects such as weak anchoring (normal and tangential), as well as fully coupled electro-statics with flexo-electric and order-electric effects. We also present several simulations (in 2-D and 3-D) illustrating the effects of the different elastic constants and electric field parameters.

Modelling and computation of liquid crystals

“…They also proved Γ-convergence of discrete global minimizers to continuous ones as the mesh size goes to zero. A similar idea has been applied to the generalized Ericksen model with eight independent 'elastic' constants (Walker 2020) and a uniaxially constrained Q-tensor model (Borthagaray, Nochetto and Walker 2020 For the full Q-tensor model, Gartland Jr et al (1991) constructed a numerical procedure that minimizes the LdG free-energy model, which is based on a finite-element discretization to the tensor order parameter, and a direct minimization scheme based on Newton's method and successive over-relaxation. The corresponding analytical and numerical issues of this numerical procedure were addressed by Davis and Gartland Jr (1998), who proved the well-posedness of the discrete problem.…”

Mathematical problems of nematic liquid crystals: between dynamical and stationary problems

“…For the QS model, some results are available in [49][50][51][52]. The E system is studied analytically in [28] and numerically in [53,54].…”

Modeling and Computation of Liquid Crystals

“…They also prove Γ-convergence of discrete global minimizers to continuous ones as the mesh size goes to zero. Similar idea has also been applied to generalized Ericksen model with eight independent "elastic" constants (Walker 2020) and an uniaxially constrained Q-tensor model (Borthagaray, Nochetto and Walker 2019).…”